Adiabatic quantum computing (AQC) can be protected against thermal excitations via an encoding into error detecting codes, supplemented with an energy penalty formed from a sum of commuting Hamiltonian terms. Earlier work showed that it is possible to suppress the initial thermally induced excitation out of the encoded ground state, in the case of local Markovian environments, by using an energy penalty strength that grows only logarithmically in the system size, at a fixed temperature. The question of whether this result applies beyond the initial time was left open. Here we answer this in the affirmative. We show that thermal excitations out of the encoded ground state can be suppressed at arbitrary times under the additional assumption that the total evolution time is polynomial in the system size. Thus, computational problems that can be solved efficiently using AQC in a closed system setting, can still be solved efficiently subject to coupling to a thermal environment. Our construction uses stabilizer subspace codes, which require at least $4$-local interactions to achieve this result.
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